Computational Science is doing Science via Computation.
It has now become the third mode of scientific research,
complementing Theory and Experiment.
Thanks to the tremendous advances in computing power,
increasingly complicated and realistic processes can now be
simulated computationally.
The design, prototyping, optimization, and control of technological
processes depend on fundamental understanding of the phenomena involved,
their interactions, and sensitivity on parameters.
These can be studied very effectively via computer simulations
based on mathematical models expressing physical principles.
- Modeling:
The first step is the "mathematization" of the process,
namely the development of a mathematical model
of the physical process. It is often by far the most
difficult.
- Analysis:
Then the full arsenal of mathematical tools can be used to
analyze and understand the basic properties of the mathematical
model and attempt to predict its behavior.
This is where theoretical issues (existence, uniqueness,
stability of solutions) become important, often generating
challenging problems for theoretical mathematical research.
- Scientific Computing
comes into play next, in developing appropriate and effective
numerical methods for the mathematical problem.
- Programing:
The algorithms are implemented in a high level computer
language (Fortran, C, C++).
- Verification:
The computer code is tested extensively on
problems/processes with known solution/behavior.
Bugs are corrected and the code is tested over and over.
- Optimization:
The computer code is optimized for robustness, stability,
and efficiency.
- Parallelization:
Realistic problems are typically very demanding computationally,
so parallelization of the code is necessary, to run on
clusters of multiprocessors and/or many networked computers
simultaneously.
- Validation:
The final crucial step is validation of the model, by simulating
a process and comparing the results of numerical simulations
with experimental measurements. It may require several
iterations of all the steps above !
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